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If / then questions part 1

q) Suppose the following statement is true: If it is Monday, then Bob goes to the park. Which of the following must also be true?

A) If it is not Monday, then Bob does not go to the park

B) If Bob goes to the park, then it is Monday

C) Bob’s favorite place is the park

D) If Bob doesn’t go to the park, then it is not Monday

E) Bob doesn’t like going to the beach

We call this type of question an if/ then question. Barron’s has a pretty solid section on these questions, but since this is one of the harder question types I find it helpful to go over them in a little more depth. You should definitely check out this section of Regents Prep, and if you have the time and patience to go really deep, this section can help make you a Logical Reasoning wiz!

Note: At least in my edition of the Barron’s book, one or two questions in the if/then section have more than one correct answer. Don’t let that confuse you!

Using Symbols

Now lets dive into the example:

It is helpful to write down if/ then statements in shorthand. The statement “If it is Monday, then Bob goes to the park” is made up of two sub-statements:

It is Monday

Bob goes to the park

Let’s let the letter M stand for “it is monday” and P stand for Bob goes to the park. The symbol stands for “If… then…”.

So we can write “If it is Monday, then Bob goes to the park” in symbols as . You say this aloud as “If M then P”. Sometimes I like to say something like “If Monday, then Park”.

So what does this if/ then statement tell us? Well, we know if it’s Monday, then Bob must go to the park.

What if it’s not Monday? You might think “If it’s not Monday, then Bob doesn’t go to the park”. That’s even an answer choice!

And that choice is...

“If it is Monday, then Bob goes to the park” only tells us about what happens when it’s Monday! It doesn’t say anything about what happens when it’s not Monday. Choice A is wrong because we don’t have any information about what happens when it’s not Monday! Maybe Bob goes to the park, maybe he doesn’t; we have no idea.

There is even a special name for this wrong answer choice: It’s called the inverse!

Negation

Before we can understand the inverse, we have to understand negation: the opposite of a statement is called the negation. Negations are often created by adding a not. For example, the negation of “It is Monday” would be “It is not Monday”. The symbol for “not” is . So if stand for “It is Monday”, then “It is not Monday” would be . You say as “not M”.

Now you try:

q) What is the negation of “Bob goes to the park”?

Solution

The negation of “Bob goes to the park” is “Bob does not go to the park”. Since stands for “Bob goes to the park”, stands for “Bob does not go to the park”.

A twist:

q) What is the negation of “It is not raining”?

Solution

“It is not raining” is already the negation of “It is raining”. Let’s say stands for “It is raining”. Then “It is not raining” would be . The negation of “It is not raining” would be “It is not not raining” or . But that means the same thing as “It is raining”. So the negation of “It is not raining” is “It is raining”. In symbols, is the same thing as .

The Inverse- wrong answer

So what does negation have to do with the inverse? Well, to get the inverse of an if/ then, you replace the if and the then parts with their negations. For example, the inverse of “If it is Monday, then Bob goes to the park” would be “If it is not Monday, then Bob does not go to the park”. Symbolically, the inverse of is .

Let’s practice getting the inverse of a couple if/ then statements:

q) What is the inverse of “If it is warm out, then Jane goes running”?

Solution

Negate both statements. The inverse is “If it is not warm out, then Jane does not go running”.

q) What is the inverse of “If it is snowing, then the pool is not open”?

Solution

Negate both statements. The second statement, “the pool is not open”, is already negated, so we un-negate it. The inverse is “If it is not snowing, then the pool is open”.

Remember the inverse of an if/then is not necessarily true and is a wrong answer choice. If we are given “If it is Monday, then Bob goes to the park”, we don’t know what happens if it’s not Monday. We can’t say “If it is not Monday, then Bob does not go to the park”. That would be a wrong answer choice.

What if we knew that Bob went to the park? Does that mean it has to be Monday? Can we choose choice B, “If Bob goes to the park, then it is Monday”?

Guess what?

We can’t say for sure that if Bob went to the park, it must be Monday. We don’t know what happens when it’s not Monday! It’s possible he goes to the park when it’s not Monday also, so just because he went to the park doesn’t necessarily mean it must be Monday.

The Converse- wrong answer

Choice B is also a special type of wrong answer. It is called the converse. To get the converse, you switch the if and the then. In this example, we started with “If it is Monday, then Bob goes to the park”, so the converse would be “If Bob goes to the park, then it is Monday”. In symbols, we started with , and the converse is .

Let’s practice finding the converse.

q) “If Samuel L. Jackson is in a movie, then the movie is good”

Solution

Switch the if and the then. The converse is: “If a movie is good, then Samuel L. Jackson is in it.”

q) “If it is Saturday, then school is closed”

Solution

Let’s try this one with symbols. Let’s say means “it is Saturday” and means “school is closed”. Then the original statement would be . So if we switch the if and the then, we get , or “If school is closed then it is Saturday”. Note that, in the real world, it is usually true that “If it is Saturday then school is closed”, but “If school is closed then it is Saturday” is not necessarily true because school could also be closed on Sunday, or a holiday, or during the summer. We don’t want to bring outside knowledge to these questions, but this example helps to show how the converse of a true statement doesn’t have to be true.

The Contrapositive- right answer

So we know that the inverse and the converse are often wrong answers. Let’s talk about one more scenario. We are starting with “If it is Monday, then Bob goes to the park”. What if we know that Bob didn’t go to the park today? Then it can’t be Monday, because on Monday Bob goes to the park! So we can say “If Bob does not go to the park, then it is not Monday”. This is called the contrapositive of the original statement, and it is the right answer! We see choice D is the contrapositive (choices C and E we don’t have information about- they’re kind of nonsense).

We can think of the if part of the if/ then as the cause and the then part as the effect. So we can restate all if/ then’s as “If the cause happens, then the effect happens”. The contrapositive is saying “If the effect didn’t happen, then the cause couldn’t have happened”. The cause forces the effect to happen.

We often have to figure out the contrapositive in these questions. To make the contrapositive, you switch the if and the then and negate them both. So the contrapositive of “If it is Monday, the Bob goes to the park” is “If Bob does not go to the park, then it is not Monday”. In symbols, the contrapositive of is .

Try making contrapositives on your own:

q) If you work, then you get paid.

Solution

If you don’t get paid, then you don’t work.

q) All purple people are not ticklish.

Solution

This is not in if/then form. We have to change it to if/then form before we do anything else. The if/then form would be “If a person is purple, then they are not ticklish”. We then switch and negate the if and then then to get the contrapositive: “If a person is ticklish, then they are not purple”. The SHSAT very often has if/then statements that are not quite in if/then form. You should immediately change them to if/then form when you see them.

q) If you don’t study for the test, you won’t pass

Solution

First, put this in full if/then form: “If you don’t study for the test, then you won’t pass”. The contrapositive is “If you pass the test, then you studied”.

Examples

Now let’s try a couple SHSAT style if/then questions.

q) Suppose the following statement is true: If Alison plays tennis, then Ryan does not play football. Which of the following must also be true?

a) If Ryan does not play football, then Alison plays tennis

b) Alison likes tennis more than football

c) If Ryan plays football, then Alison does not play tennis

d) If Ryan plays tennis, then Alison plays football

e) If Alison does not play tennis, then Ryan plays football

Solution

First let’s get the contrapositive. Let T = Alison plays tennis and F = Ryan plays football. Then the original statement is . So we switch the if and then and negate them both to get the contrapositive, which is: . In English this is “If Ryan plays football, then Alison does not play tennis”. Now let’s look through the answer choices:

a) If Ryan does not play football, then Alison plays tennis Converse- wrong

b) Alison likes tennis more than football Not enough information about this- wrong

c) If Ryan plays football, then Alison does not play tennis Contrapositive- right

d) If Ryan plays tennis, then Alison plays football We aren’t told anything about Ryan playing tennis or Alison playing football-wrong

e) If Alison does not play tennis, then Ryan plays football Inverse-wrong

So choice C, the contrapositive, is correct.

q) Suppose the following statement is true: Joan does not wake up early if school is not open. Which of the following must also be true?

a) If school is open, then Joan wakes up early.

b) Joan is a heavy sleeper.

c) If Joan wakes up early, then school is open.

d) If Joan does not wake up early, then school is not open.

e) If school is open, then Joan does not wake up early.

Solution

First we have to put the statement in regular if/then form. Joan does not wake up early if school is not open means “If school is not open, then Joan does not wake up early”. Let O mean “School is open” and W mean “Joan wakes up early”. Then the statement in symbols is . The contrapositive is , or “If Joan wakes up early, then school is open”. This is choice C!

a) If school is open, then Joan wakes up early. Inverse- wrong

b) Joan is a heavy sleeper. wrong-Not enough information about this

c) If Joan wakes up early, then school is open. Contrapositive- correct

d) If Joan does not wake up early, then school is not open. Converse- wrong

e) If school is open, then Joan does not wake up early. wrong- we don’t know what happens if school is open